In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups
The direct sum decomposition is usually referred to as gradation or grading.
A graded module is defined similarly (see below for the precise definition).
It generalizes graded vector spaces.
The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.
Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified.
A graded ring is a ring that is decomposed into a direct sum of additive groups, such that for all nonnegative integers
By definition of a direct sum, every nonzero element
called the homogeneous part of degree
Serre's twisting sheaf in algebraic geometry).
Given a graded module M over a commutative graded ring R, one can associate the formal power series
It is called the Hilbert–Poincaré series of M. A graded module is said to be finitely generated if the underlying module is finitely generated.
, k a field, and M a finitely generated graded module over it.
is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M. An associative algebra A over a ring R is a graded algebra if it is graded as a ring.
In the usual case where the ring R is not graded (in particular if R is a field), it is given the trivial grading (every element of R is of degree 0).
One example is the close relationship between homogeneous polynomials and projective varieties (cf.
The above definitions have been generalized to rings graded using any monoid G as an index set.
A G-graded ring R is a ring with a direct sum decomposition such that Elements of R that lie inside
The previously defined notion of "graded ring" now becomes the same thing as an
is the monoid of natural numbers under addition.
The definitions for graded modules and algebras can also be extended this way replacing the indexing set
with any monoid G. Remarks: Examples: Some graded rings (or algebras) are endowed with an anticommutative structure.
Specifically, a signed monoid consists of a pair
That is, the set of elements of the graded monoid is
where g is the cardinality of a generating set G of the monoid.
That is, there is no unit divisor in such a graded monoid.
, the indexing family could be any graded monoid, assuming that the number of elements of degree n is finite, for each integer n. More formally, let
denotes the semiring of power series with coefficients in K indexed by R. Its elements are functions from R to K. The sum of two elements
is defined pointwise, it is the function sending
This sum is correctly defined (i.e., finite) because, for each m, there are only a finite number of pairs (p, q) such that pq = m. In formal language theory, given an alphabet A, the free monoid of words over A can be considered as a graded monoid, where the gradation of a word is its length.